If Doodle told the story, it would change the focus of the narrative. We would focus on Doodle's thoughts and feelings instead of those of his brother. If Doodle told his own story, we would be able to get very close to him. As it is, we can only read the details of the story and infer what Doodle is thinking and feeling.
If Doodle were the narrator, we would hear in his own words how much he loved his brother, why he loved him so much, how hard Doodle tried to please him, and how difficult it was to try. Doodle would explain to us why he loved the scarlet ibis and wanted to bury it. He would talk about how much he loved the natural world when he was able to go out into it. As a narrator, Doodle would be able to express all of his emotions.
We might ask why the author did not make Doodle the narrator. The author chose to make this Doodle's brother's story to develop the theme of pride and its consequences. Doodle's brother learns a very hard lesson.
If the story were from Doodle's point of view, then the action would most likely start later. Readers would have to wait until Doodle could form sentences and express himself before hearing his perspective on the world. As Doodle doesn't live that long, this would greatly shorten up the timeline of the plot.
However, readers would be able to see the brother from Doodle's eyes. The brother comes off as remorseful and readers have sympathy for him. Doodle's perspective might increase the sympathy by showing love for the brother, or it might take away the sympathy by showing the brother as a bully. At any rate, readers would be able to understand what Doodle suffers through in his short life as a crippled boy. The theme would revolve around the treatment of the handicapped and how family can either be an aid or a hindrance. There might be a theme related to determination as we listen to Doodle struggle and cheer himself on.
The ending would likely be more intense and perhaps more disturbing. To be in Doodle's frame of mind when he is left alone in the storm would be to change the story from a drama to a horror story.